Flexoelectricity and three-dimensional solitons in nematic liquid crystals

University of Minnesota Ph.D. dissertation.December 2019. Major: Mathematics. Advisor: Maria-Carme Calderer. 1 computer file (PDF); vi, 103 pages.We model a novel type of three-dimensional, chevron soliton found in nematic electrolytes with negative dielectric and conducting anisotropies, subject to an alternating electric (AC) field \cite{Tuxedos}. Flexoelectricity couples electric field to splay and bend distortions in the nematic, and was proposed as the driving mechanism of the solitons in \cite{Tuxedos}. In this work, we first derive dynamic equations for a flexoelectric liquid crystal with ionic impurities. We then linearize about the initial uniform state, find that it is unstable to soliton-like perturbations, and recover estimates for the size of the traveling wave. In this work we also study the equilibrium problem of a flexoelectric liquid crystal without ionic effects, and prove well-posedness of both systems